Realizing cyclic linear transformations as Frobenius elements in the Galois groups of $q$-polynomials over function fields

TL;DR

This paper demonstrates how to realize Frobenius conjugacy classes and transvections in Galois groups of certain $q$-polynomials over function fields, enabling identification of these groups as classical groups.

Contribution

It introduces a method to realize specific elements in Galois groups of $q$-polynomials, aiding in their classification as classical groups.

Findings

Frobenius conjugacy classes are realized using degree 1 ideals.

Transvections are constructed in linear Galois groups.

Galois groups are identified as classical groups in various cases.

Abstract

We realize Frobenius conjugacy classes in Galois groups of certain $q$-polynomials over $\mathbb{F}_q(t)$ using specific degree 1 ideals. We combine this with methods from elementary linear algebra and group theory to realize transvections in some linear Galois groups. This enables the Galois group to be identified as a known classical group in several reasonably general cases.